Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com 5.5 The Expected Value of a Function of Random Variables 255 If Y 1 and Y 2 are independent, show that P(a < Y 1 ≤ b, c < Y 2 ≤ d) = P(a < Y 1 ≤ b) × P(c < Y 2 ≤ d). [Hint: Express P(a < Y 1 ≤ b) in terms of F 1 (·).] 5.68 A bus arrives at a bus stop at a uniformly distributed time over the interval 0 to 1 hour. A passenger also arrives at the bus stop at a uniformly distributed time over the interval 0 to 1 hour. Assume that the arrival times of the bus and passenger are independent of one another and that the passenger will wait for up to 1/4 hour for the bus to arrive. What is the probability that the passenger will catch the bus? [Hint: Let Y 1 denote the bus arrival time and Y 2 the passenger arrival time; determine the joint density of Y 1 and Y 2 and find P(Y 2 ≤ Y 1 ≤ Y 2 + 1/4).] 5.69 The length of life Y for fuses of a certain type is modeled by the exponential distribution, with { (1/3)e −y/3 , y > 0, f (y) = 0, elsewhere. (The measurements are in hundreds of hours.) a b If two such fuses have independent lengths of life Y 1 and Y 2 , find the joint probability density function for Y 1 and Y 2 . One fuse in part (a) is in a primary system, and the other is in a backup system that comes into use only if the primary system fails. The total effective length of life of the two fuses is then Y 1 + Y 2 . Find P(Y 1 + Y 2 ≤ 1). 5.70 A supermarket has two customers waiting to pay for their purchases at counter I and one customer waiting to pay at counter II. Let Y 1 and Y 2 denote the numbers of customers who spend more than $50 on groceries at the respective counters. Suppose that Y 1 and Y 2 are independent binomial random variables, with the probability that a customer at counter I will spend more than $50 equal to .2 and the probability that a customer at counter II will spend more than $50 equal to .3. Find the a joint probability distribution for Y 1 and Y 2 . b probability that not more than one of the three customers will spend more than $50. 5.71 Two telephone calls come into a switchboard at random times in a fixed one-hour period. Assume that the calls are made independently of one another. What is the probability that the calls are made a b in the first half hour? within five minutes of each other? 5.5 The Expected Value of a Function of Random Variables You need only construct the multivariate analogue to the univariate situation to justify the following definition.